Major General Joseph Votel of US special operations command has been appointed to head the investigation into the death of the British aid worker Linda Norgrove, US central command announced today.

Britain is to appoint Brigadier Rob Nitsch of the joint support force in Kabul – the head of logistics for UK forces in Afghanistan – to "work closely" with the US general.

Last night, Barack Obama phoned Downing Street to promise to "get to the bottom" of Friday's failed rescue attempt in Afghanistan's eastern Kunar province. Video evidence suggested Norgrove was killed by an American grenade, and not by her captors as the US had claimed over the weekend, and a full investigation has been promised.

"The teams are discussing precisely how that work will take place and the detailed terms of reference," said a Downing Street spokesman.

Commander General James Mattis, of the US army, will discuss the investigation with David Cameron's national security adviser, Sir Peter Ricketts, during a visit to London tomorrow.

The announcement comes after Obama spoke with Cameron last night.

"The president offered his condolences for the death of Linda Norgrove," a Downing Street spokesman said. "They agreed that the decision to launch the rescue operation had been right, given the grave danger to Linda's life, and that US forces had shown great courage.

"The president agreed that it was now essential to get to the bottom of what had happened in the course of the rescue operation. They looked forward to close co-operation between the UK and US authorities on the investigation and agreed to stay in close touch as it moved forward."

Richard Ottaway, the Conservative MP and chairman of the foreign affairs select committee, said the joint inquiry would need to ask why US forces used live grenades in the raid.

He said at a press conference that earlier reports of how she had died, which were repeated by the foreign secretary, William Hague, were "deeply regrettable" but had been made in good faith.

Norgrove, 36, was working with the US aid organisation Development Alternatives Inc (DAI), when she was captured at the end of September on her way to inspect an irrigation project.

Three Afghan colleagues captured with her were released unharmed a few days later.

The DAI vice-president, Betsy Marcotte, who visited Norgrove's family on the Isle of Lewis, said she did not blame US troops for her colleague's death.

Marcotte told BBC Radio 4's Today programme: "Clearly it is distressing news, but it doesn't change anything for me. I feel confident in how this was handled and that we have been satisfied that the British and the Americans were doing everything they could."

After spending the day with the Norgrove family, Marcotte added: "I really feel that they are doing remarkably well, given how difficult this is.

"It reminds me again that Linda exuded strength and was a very calm presence and very comfortable in her own being. Certainly that was something I was always impressed with when I was with her."

Marcotte said DAI had not been involved in the decision-making process ahead of the rescue attempt, but had done whatever it could to support the British and US authorities.

Sean Langan, an award-winning British documentary maker, was held hostage by Islamist extremists in Afghanistan for three months in 2008. He was released after successful negotiations with his kidnappers